Sen. Ted Cruz said Sunday that the right to bear arms and other “basic liberties” will be hanging in the balance whenever the Senate votes to confirm a replacement for Supreme Court Justice Antonin Scalia, whose sudden death Saturday upended the presidential race and set off a tenacious fight over President Obama’s right to nominate a successor.

Mr. Cruz, the Texas Republican who won the Iowa caucuses, is trying to rally conservatives ahead of the South Carolina primary on Feb. 20. He said it is too late into Mr. Obama’s term to let him nominate someone and tilt the court toward a liberal majority.

And he used the occasion to take a swipe at businessman Donald Trump, saying the GOP presidential front-runner doesn’t have the conservative bona fides to select the right justice, if Republicans fulfill their pledge to thwart Mr. Obama.

“We’re one justice away from the Second Amendment being written out,” Mr. Cruz told ABC’s “This Week.” “It is abundantly clear that Donald Trump is not a conservative, he will not invest the capital to confirm a conservative.”

Mr. Cruz and other GOP rivals used a Saturday-night debate in South Carolina to paint Mr. Trump as a nascent conservative, pointing to his coziness with Democrats in the past and his prior support for abortion rights and a single-payer health care system.

Mr. Trump hit back hard, saying Mr. Cruz supported Chief Justice John G. Roberts Jr. only to see him uphold Obamacare in two major opinions.

“He stands on the Senate floor, he’s got no support from one senator,” Mr. Trump told ABC, falling back on his favored attack on Mr. Cruz. “You look at his colleagues, he has absolutely no endorsements. He has no support. He’s a lone wolf and he’s going to get nothing done. He’s not a leader.”

Mr. Trump then trained his fire on Jeb Bush, the former Florida governor who is hoping to get a boost from his brother — former President George W. Bush — ahead of the South Carolina primary.

The real estate mogul said the former president made a grave mistake by attacking Iraq, and that his brother Jeb stumbled out of the gate when he was asked if the war had been a mistake.

“His pollsters ultimately gave an answer five days late. He was mumbling back and forth, you remember that fiasco. He was gone by the time he started,” Mr. Trump said. “When he announced, he was practically done. He couldn’t even answer whether Iraq was a good thing or a bad thing. Ultimately, he determined that the Iraq war was a bad thing.”